Asynchronous to synchronous sampling using an augmented least squares solver

ABSTRACT

A method, comprising: receiving a plurality of 2-tuples of asynchronously sampled inputs at an asynchronous to synchronous reconstructor; performing a coarse asynchronous to synchronous conversion using the plurality of 2-tuples to generate a plurality of low precision synchronous outputs; generating a high precision synchronous output, z 0 , using a plurality of asynchronous 2-tuples, low precision synchronous outputs after it, and its own high precision outputs from previous steps; calculating c 0  and c −1  by summing future low precision outputs and the past high precision outputs after they are weighted with the appropriate windowed sinc. values and then subtracted from appropriate asynchronous samples; calculating, the four quantities “s −11 ”, “s 01 ”, “s 00 ” and “s −10 ” based on particular values of the windowed sinc. function; and using c 0 , c −1 , s −11 , s 01 , s 00  and s −10 , the high precision synchronous output of interest, z 0  is generated.

PRIORITY

This application claims priority to Indian Application No.1066/CHE/2013, filed Mar. 13, 2013, entitled “Leap Frog Sampler forAsynchronous ADCs”, Indian Provisional Application No. 971/CHE/2013,filed Mar. 6, 2013, entitled “Low Complexity Non-Uniform InterpolationAlgorithm”, Indian Application 927/CHE/2013, filed Mar. 4, 2013,entitled “High Performance Non-Uniform Interpolation”, IndianApplication 913/CHE/2013 “ADC Range Extension Using Post-ProcessingLogic” filed Mar. 1, 2013, and Indian Application No. 1132/CHE/2013,entitled “Rate Enhancement Techniques for Asynchronous Samplers”, filedMar. 15, 2013, the entireties of all of which are hereby incorporated byreference. Also, this application claims the benefit of U.S. ProvisionalApplication No. 61/922,271, filed Dec. 31, 2013, and U.S. ProvisionalApplication No. 61/922,282, filed Dec. 31, 2013, and U.S. ProvisionalApplication No. 61/922,291, filed Dec. 31, 2013, and U.S. ProvisionalApplication No. 61/922,309, filed Dec. 31, 2013, and U.S. ProvisionalApplication No. 61/922,316, filed Dec. 31, 2013, and U.S. ProvisionalApplication No. 61/922,533, filed Dec. 31, 2013, the entireties of allof which are hereby incorporated by reference.

TECHNICAL FIELD

This application is directed, in general, to an analog to digitalconverter (ADC), and more particularly, to an asynchronous ADC thatreduces the excessive oversampling, inherent in level crossing baseddata converters, and reduces the complexity of reconstruction ofasynchronously-sampled signals.

BACKGROUND

Generally, there are two basic types of ADC converters: a) synchronousand b) asynchronous. Synchronous ADCs are sampled at fixed intervals, asin FIG. 1A; and asynchronous ADCs, such as FIG. 1B, are sampled atvariable intervals. Both types of ADCs have their weaknesses.

With synchronous ADCs, such as in FIG. 1A, there is the problem with its“quantization error” regarding measuring of analog input signals. Eg. ifan input signal is between levels to be sampled by the ADC, when asampling is “forced” by a clock, the digital sample, which coincides toone the ADC levels surrounding the signal, is generated. At thisinstant, a small error (voltage quantization error) is introduced in tothe system, and a small fraction of the signal information is “lost”.

Moreover, regarding synchronous ADCs, whether the input is full scale orhalf-full scale or one-tenth full-scale, the amount of quantizationnoise is actually the same; therefore, as the signal becomes smaller,the ratio of signal energy to quantization noise energy actually getsproportionately smaller.

One way of expressing a signal to noise ratio, is that of “EffectiveNumber of Bits” or “ENOB”. ENOB expresses the signal to noise ratio interms of bits. As the input amplitude goes down, and the noise amplitudedoes not go down, then the ENOB also goes down—in other words, theprecision of the digital output drops as the analog input signalamplitude drops since the noise amplitude holds steady.

In a synchronous ADC, if the signal level decreases, the noise[quantization error] remains the same, so there is a loss of signal tonoise ratio. If quantization noise is the only noise source,specifically signal to quantization noise ratio (SQNR) decreases. Inreality, the noise has multiple components such as thermal noise,quantization noise, noise due to non-linearity of devices, etc. Even inthis case with all noise sources, SNR decreases as signal amplitude goesdown or any of the noise sources increase.

There is an alternative approach to analog to digital conversion, usingwhat is known as an “asynchronous analog to digital converter”(“asynchronous ADC”), such as in FIG. 1B. With asynchronous converters,such as discussed in “A non-uniform sampling technique for A/Dconversion, Circuits and Systems, 1993, ISCAS '93, 1993 IEEEInternational Symposium on, vol., no., pp. 1220-1223 vol. 2, 3-6 May1993”, by Sayiner, N.; Sorensen, H. V.; Viswanathan, T. R. (“Sayiner”);a digital sample is generated when the input analog signal crosses athreshold (“level”). This is advantageous, because there is no voltagequantization error when signal crosses a known “level”. This means thatthere is no voltage quantization noise in the output samples from anasynchronous ADC by itself. Please see FIGS. 1A-2B regarding thequantization noise, as shall be explained, below.

Moreover, the output of the asynchronous ADC is not directly usable bysynchronous signal processing circuits that follow the ADC. Therefore,at the time of the conversion, a “2-tuple” is generated upon crossing athreshold, that of an output level and a “timestamp”, which may be usedin further processing to create synchronous samples usable by standardsynchronous signal processing circuits

However, the timestamp itself has quantization noise, a “time”quantization noise. This time quantization step and the associatedeffects are illustrated in FIG. 2A. Let us assume that the signalcrosses level V₂, at time “T”. Now, ideally [V₂, T] would be theasynchronous output. To reduce complexity the time “T” is stored inquantized form—chosen from the edges of a high resolution clock (T₁, T₂,T₃ . . . ). So, instead of [V₂, T], [V₂, T₂] is the 2-tuple actuallystored, and hence a time quantization error ΔT=|T₂-T| is introduced.

As these time grids are assumed to be finely spaced from each other, thesignal in this region can be approximated by a straight line with slopedV/dT. Hence the voltage quantization error, ΔV, introduced due to thistime quantization error can be approximated as ΔV=dV/dT. ΔT. This showsthat this kind of quantization error is proportional to signal, andhence the Signal to quantization noise from such quantization noisesources hold constant even if signal energy drops. In systems where thisnoise source is dominant, the SNR (or equivalently ENOB) does not dropwith decrease in signal level.

Now, a given system may have different noise sources—elastic noisesources (like SNQR due to time quantization, jitter noise, etc.) whichscale with signal levels, and inelastic noise sources (like voltagequantization noise present in synchronous ADCs, thermal noise, flickernoise, etc.) which do not scale with signals. If the dominant noisesource is elastic in nature, the SNR (or ENOB) does not fall even ifsignal amplitude falls. Asynchronous systems can be designed such thatinelastic noise sources become dominant only beyond a point—and tillthen the SNR does not fall even if signal falls. This region where theSNRs hold steady, is called a “flat band”—after which the inelasticnoise sources become dominant, and the ENOB starts falling (similar tosynchronous ADCs) as signal levels fall.

Moreover, regarding SQNR, as is understood by the present inventors,SQNR is proportional to the value computed by the expression, “−log(bandwidth*time quantization error)”. As the bandwidth of theasynchronous ADC goes down, and the SQNR improves, for a given timequantization error.

FIG. 2B illustrates a “flat band” for an asynchronous ADC converter. Asis illustrated, a hypothetical customer requirement is proposed. Whenthe customer seeks 11.5 effective bits at ¼ fullscale, a conventionalADC provides that using a 14 bit (13.5 effective bits) ADC whereas theasynchronous ADC (with a 2-bit flat band) can provide the same with 12effective bits ADC. Typically lower bit requirements translate toexponentially lower power requirements.

The red line shows the decrease in ENOB for a 12 effective bitssynchronous ADC, which provides only 10 effective bits at ¼ full scale,thereby not satisfying the customers' requirements. Furthermore,traditional asynchronous ADC architectures, however, each have their owndrawbacks.

As alluded to above, for conventional asynchronous ADC, the samplingrate is a function of the signal amplitude, number of active levels usedfor level crossing detection and signal frequency. Moreover, for somesignal types, classical asynchronous ADC, as reported in Sayiner's work,tend to oversample the signal. For e.g., for an 8 level asynchronousADC, a full-scale sine wave is sampled sixteen times, which is 8-timesthe minimum samples (“Nyquist rate”) required for signal reconstruction.A 16 level asynchronous ADC may sample the same signal thirty two times,which is sixteen times more than the minimum required. Therefore, in theprior art, one always have an extremely large amount of oversampling.

That in turn translates to large amounts of power wasted especially ifmost of the samples are thrown away by the signal processing chainfollowing the ADC. In addition, another drawback of traditionalasynchronous ADCs is that they produce asynchronous (non-uniformlyspaced in time) samples. Since the domain of asynchronous signalprocessing is nascent, most applications prefer to process synchronoussamples. However, converting these non-uniform samples into uniformsamples is a complex and power hungry task.

Furthermore, when converting, it would be advantageous to have anasynchronous ADC that addresses at least some of these drawbacks.

SUMMARY

A first aspect can provide a method, comprising: receiving an analoginput; determining an upper outer rail and a lower outer rail as pollingvalues to be used by voltage comparators; blanking at least threecomparators; determining which two of the at least three comparators areclosest to the input analog voltage levels; defining the two comparatorswhich are closest to the analog input signal to be the set of railcomparators of the next sampling process; assigning a remainingcomparator at a voltage level in between the new top and bottom voltagelevels; enabling the outer rails, but blanking the other comparator;progressively narrowing down the voltage range spanned by the two outercomparators and finally generating a 2-tuple value of an asynchronousvoltage comparator crossing.

A second aspect includes an apparatus, comprising: means for receivingan analog input; means for determining an upper outer rail and a lowerouter rail as polling values to be used by voltage comparators; meansfor blanking at least three comparators; means for determining which twoof the at least three comparators are closest to the input analogvoltage levels; means for defining the two comparators which are closestto the analog input signal to be the set of rail comparators of the nextsampling process; means for assigning the remaining comparator at avoltage level in between the new top and bottom voltage levels; meansfor enabling the outer rails, but blanking the other comparator; meansfor progressively narrowing down the voltage range spanned by the twoouter comparators and finally means for generating a 2-tuple value of anasynchronous voltage comparator crossing.

A third aspect provides an apparatus, comprising: an asynchronoussampler, including: at least three voltage comparators, wherein thevoltage comparators are configured to: receive an analog input;determine an upper outer rail and a lower outer rail as polling valuesto be used by voltage comparators; blank at least three comparators;determine which two of the at least three comparators are closest to theinput analog voltage levels; define the two comparators which areclosest to the analog input signal to be the set of rail comparators ofthe next sampling process; assign the remaining comparator at a voltagelevel in between the new top and bottom voltage levels enable the outerrails, but blanking the other comparator; progressively narrow down thevoltage range spanned by the two outer comparators and finally generatea value of an asynchronous voltage comparator crossing; a time stamper,when the output of this is combined with the asynchronous sampler,creates a 2-tuple; and a digital reconstructor and resampler thatcreates a synchronous sampled signal from the asynchronously sampledsignal.

BRIEF DESCRIPTION OF THE DRAWINGS

Reference is now made to the following descriptions:

FIG. 1A illustrates a prior art synchronous ADC sampling;

FIG. 1B illustrates a prior art asynchronous ADC sampling;

FIG. 2A illustrates an effect of time quantization as it relates to anequivalent voltage quantization as understood by the present inventors;

FIG. 2B illustrates a flat band of a 13.5 effective bit asynchronousADC;

FIG. 3A illustrates an asynchronous ADC constructed according to theprinciples of the present application.

FIG. 3B illustrates a table of acronyms and other terms in the followingdiscussion;

FIG. 4 illustrates a scheme for setting voltage “rail” voltage levelswithin an asynchronous ADC according to the principles of the presentapplication;

FIGS. 5 i and 5 ii illustrates a method for setting voltage “polling”voltage levels with the asynchronous ADC of FIG. 3A;

FIG. 6A illustrates a block diagram of an asynchronous ADC designedaccording to the principles of the present application;

FIG. 6B illustrates a snapback selector of the snapout controller ofFIG. 6A constructed according to the principles of the presentapplication;

FIG. 7 illustrates a first example method flow for determining a reducedsnapout range for an asynchronous ADC constructed according to theprinciples of the present application;

FIG. 8 illustrates a second example method flow for determining areduced snapout range for an asynchronous ADC constructed according tothe principles of the present application;

FIG. 9 illustrates the predictor snapout calculator of FIG. 6 in moredetail;

FIG. 10A illustrates an example asynchronous to synchronous calculatorsof the synchronous digital reconstructor of FIG. 3A;

FIG. 10B illustrates a graphed output of an Akima algorithm and a“modified Akima algorithm” that calculates an analog-to digitalsynchronous signal from the inputted sampled asynchronous signals ascalculated in the synchronous digital reconstructor of FIG. 3A;

FIG. 11 illustrates an example method for calculating synchronoussamples from asynchronous samples according to the Akima approach ascalculated in the synchronous digital reconstructor 308 of FIG. 3A;

FIG. 12 illustrates an example method for calculating synchronoussamples from asynchronous samples according to the modified Akimaapproach as calculated in the synchronous digital reconstructor of FIG.3A;

FIG. 13 illustrates a Table that shows the improvements that occur withemployment of the Akima and “modified Akima” algorithm for determiningsynchronous samples from asynchronous samples;

FIG. 14A illustrates an augmented least squares (“LS”) solverconstructed according to the principles of the present application,using a generic coarse reconstruction scheme for the first step;

FIG. 14B illustrates an augmented LS solver constructed according to theprinciples of the present application, using a modified Akima algorithm;

FIGS. 15A and 15B illustrate prior art reconstruction using a sincfunction;

FIG. 16 is a waveform that illustrates the relationships of samples ofan asynchronous ADC sampler using the augmented sampler of FIG. 14 a andFIG. 14 b;

FIG. 17 is a table of the computational complexity of variousreconstruction algorithms per output point;

FIG. 18 is a graph of merit of the reconstruction scheme used as part ofasynchronous ADC.

FIG. 19 is a method of an augmented least squares solver for generatingsynchronous data points from an asynchronous signal constructedaccording to the principles of the present application; and

DETAILED DESCRIPTION

FIG. 3A illustrates a block diagram of an example asynchronous analog todigital converter (ADC) 300 constructed according to the principles ofthe present application.

As is illustrated, the asynchronous ADC is preceded by an anti-aliasingfilter 302, which receives an analog input. The output of theanti-aliasing filter 302 is coupled to an input of an asynchronoussampler 304. The asynchronous sampler 304 output is time-stamped by thetime-stamper block which can be calibrated using an external clock, 306.The time stamped asynchronous outputs (2-tuples) are fed to a digitalreconstructor or resampler (“resampler”) 308. The resampler 308 receivesthe asynchronous samples (2-tuples) and creates synchronous digitalsamples from them.

The asynchronous sampler 304 further includes a polling referenceadaptation algorithm 310 which describes a scheme to generateasynchronous samples from a continuous analog signal.

The asynchronous sampler 304 also includes a sample rate control unit320, which controls the rate of the asynchronous samples. The samplerate is adapted such that it allows for a simplification of areconstruction algorithm used in the resampler 308. Generally, theasynchronous sampler 304 gives out the asynchronous sample 2-tuples(i.e. a voltage magnitude (level) and a time stamp measurementcorresponding to that level crossing).

The sample rate control unit 320 performs this task by performingvarious predictions as to where the various polling levels of theasynchronous ADC should be placed, that is less than the full range, asshall be described below. By placing the snapout at less than the fullrange, advantageously the ADC converter 300 can converge to a solutionor a timeout in a substantially shorter time.

The digital reconstructor and sampler 308 includes an Akimareconstruction algorithm 325 based on the algorithm described in Akima,H.; “A new method of interpolation and smooth curve fitting based onlocal procedures, Journal of the Association of Computing Machinery,Vol. 17, No. 4, October 1970, pp. 589-602” which is hereby incorporatedby reference in its entirety, a modified Akima reconstruction algorithm330 and an advanced reconstruction algorithm 340, wherein either theAkima algorithm 325, the modified Akima algorithm 330 or the advancedreconstruction algorithm 340 processes an output from the asynchronoussampler 304. These algorithms are used in the digital reconstructor 308to generate synchronous digital samples from the asynchronously sampledinput signals.

Generally, the present architecture of the asynchronous ADC 300 has beendirected towards mitigating two shortfalls of traditional asynchronousADCs—unwanted oversampling and complex (power hungry) reconstructionfrom asynchronous 2-tuples to synchronous samples.

Mathematically, it may be shown that for a perfect construction of asignal from sampled data, one needs, on an average, the Nyquist samplingrate of 2B samples per second, where B is the bandwidth of the signalbeing sampled. These 2B samples may be created synchronously orasynchronously. Therefore, any sampling that occurs more than 2B timesper second is an unwanted oversampling.

To convert from asynchronous samples to synchronous samples, there is aninterpolation step, wherein performance/power of which also depends onthe amount of samples per second. A purpose of the resampling thatoccurs with this asynchronous ADC 300 is directed towards satisfying twomajor areas, while minimizing overall power: a) the sampling shouldideally be reduced to a minimum, which in this case is as close to 2Btimes per second as possible, and secondly the asynchronous samplingshould be such that the accuracy obtained after the asynchronous tosynchronous conversion meets the ENOB requirements. In traditionalasynchronous ADCs, the more analog input levels that are used by thesampler, the more digital samples that are generated

FIG. 3B illustrates a table of acronyms and other terms in the followingdiscussion.

FIG. 4 illustrates a new approach to asynchronous analog conversion,—itis a binary search algorithm similar to a SAR. This approach ismanifested in the dynamic polling reference changer 310.

In FIG. 4, the x-axis is time and the y-axis is reference levels. FIG. 4illustrates a 3 comparator search strategy, but it can be extended tomore comparators, thereby getting a faster asynchronous rate ofsampling, at the cost of higher power. Please note that the analog inputsignal is a band-limited signal.

Except for the initial blanking period when all three comparators areinactivated at time t0 (indicated by the gray box), at all other timestwo of the comparators are “active”—i.e. their outputs are reliable andcan generate asynchronous samples if the analog input signal crossestheir reference level.

On the other hand, the third comparator (and all three comparatorsduring the initial blanking period) undergoes a reference level changeat every polling instant, and is also declared inactive (i.e. theiroutput is unreliable and is masked) during the blanking period whichfollows. Thus, the two active comparators are used to generate theasynchronous samples and the third comparator is used to help intracking the signal, as shall be discussed below.

The asynchronous samples in FIG. 4 are indicated by black dots. Everytime a black dot occurs, and the asynchronous sampling occurs—thereferences of all the comparators are changed (they are moved away fromthe signal level, i.e. “snapped” out) and then they go into the blankinginterval, so as to be able to “capture” the next asynchronous inputsignal. During this blanking time, the comparators are allowed to settleand the outputs cannot be considered reliable.

At the end of the blanking period, t1, the comparator outputs arereliable; they are used to poll the signal—to determine which halfcontains the signal.

Then the 2 outer comparators in the correct half are kept active—whilethe reference of third comparator is moved to bisect the top half to beinserted between the two levels wherein the signal lies. However, thisinterposed third comparator which has just seen its reference levelchange, is temporarily kept inactive, i.e. cannot produce samples.During this time if the signal crosses the outer levels, an asynchronoussample is created. Otherwise, after another polling period, thecomparators are flashed again and the process is repeated.

So as can be seen, after the initial blanking period, only themid-comparators are allowed to move and only its reference needs to beblanked—but the signal is always trapped by the outer level comparators.Also after every blanking period—the search space is halved by theinterposing voltage comparator so as to reduce the separation to 1 LSBin a given amount of time.

As can be imagined, a fast moving signal will cross the reference levelsquickly—while a slow moving signal might take a number of steps.Therefore, in a further aspect, the snapout after a sample can beincreased for a fast moving signal, and reduced for a slow movingsignal—to control the sampling rate.

Prediction Algorithm

The reference levels for the asynchronous sampling can indeed be drivenby a prediction algorithm, which modifies thresholds, and theprobability of such crossings, based on local characteristics of thesignal. For example, the reference levels could be placed at variousplaces other than mid-way, based on such factors as a predicted behaviorof the analog input signal, and so on.

Apart from changing the manner in which the reference levels are chosen,another way to change the sampling rate of such an ADC, is to use ahigher number of comparators. There are a number of straightforwardextensions of this scheme, where more than three comparators may be usedalong with signal slope or other parameters, so as to change thesampling rate of this ADC.

For example, if there are N comparators used, then theoretically thepolling snapout algorithm can be implemented in which the successiverange between the “rail” comparators between every polling interval isreduced by a factor (N−1) until the range reduces to “m” (where m issmall integer) LSBs. The “rail comparators” can be defined as thecomparators which are used to bound an analog signal between two levels.The “outer rail comparators” are the maximum allowable voltagecomparison levels of the asynchronous ADC 300.

Another aspect of this converter is how it handles DC or low frequencysignals. For certain such signals, even after KT seconds (where K is aconstant less than or equal to 1 and T is the Nyquist frequency of thesignal frequency)—the signal may not cross a static reference level. Toensure that the voltage quantization noise is minimum, the signal rangespanned by the outer comparators is progressively reduced over the KTseconds, till the separation between the active comparators is only 1LSB. If an asynchronous sample is not generated within this time period,a sample is forced with ½ LSB of voltage quantization noise—similar to asynchronous ADC. This is equivalent to configuring a time out, T_(out)equal to KT seconds, whereby a sample is forced at every time out if anasynchronous sample is not produced before that. The forced sample isdeemed to have voltage level equal to the mid-level between the twolevels bounding the input signal.

Escaping Signals

Also note that there is a possibility for the signal to escape duringthe initial blanking time, and in this situation, the outer comparatorsare programmed with voltage levels corresponding to one of the railvoltages of the ADC and the minimum or maximum of the current referencevoltage levels according to the direction in which signal hasescaped—and after paying the penalty of one extra polling period, thesignal can be recovered.

If the signal has escaped in the positive direction (all the comparatoroutputs are low), the upper rail of the ADC and the maximum of thecurrent reference levels are taken as the bounding reference levels. Ifthe signal has escaped in the negative direction (all the comparatoroutputs are high), the lower rail of the ADC and the minimum of thecurrent reference levels are taken as the bounding reference levels. Inthe case of an escape event, after setting the reference levels whichbound the analog input signal, the regular process of reducing thevoltage range between the rail comparators is followed until one of therail comparators is crossed or time out happens for a new samplecreation.

Timeout

To ensure that at least the minimum number of samples are produced forall signal frequencies and amplitudes, the concept of timeout is used.This ensures that the asynchronous ADC produces Nyquist rate (2B)samples for low frequency/DC signals even if it lies between twoadjacent reference (DAC) levels. The timeout is typically set at “KT”seconds where “K” is a scaling constant (K<=1) and T is the Nyquist timeinterval of the signal (T=1/(2B)). If no level crossing occurs beforethe time out period, a sample is forced in the middle of the tworeference levels which bound the input signal.

FIGS. 5Bi and 5Bii illustrates a method 500 for setting the “rail”references of the asynchronous ADC of FIG. 3A. This is a method oftracking an analog signal and generating accurate asynchronous samplesusing a multitude of comparators, each using its own dynamicallyassigned unique reference level. Each time an asynchronous sample iscreated, the outer-most comparators are widened (“snapped”) back to apredefined position, and the widening is also called “snap-out” in thisdocument.

In one aspect, the widening occurs to the maximum range allowable, oreven to the voltage rails. In another aspect, the widening could be lessthan an allowable maximum range, and could be determined based on thecharacteristics of the signal. In this document, this is referred to as“adaptive snapout”, and will be discussed in more detail below. Pleasenote that the method 500 may be read in concert with FIG. 4.

In a step 510, analog input is received by the asynchronous ADC 300.

In a step 520, the asynchronous sampler 304 of the ADC 300 determinesthe “suitable” values of outer comparators, at time t0, and innerpolling comparator level, also at t0. Suitable can be generally derivedas the extreme voltage rails, or can be based on prior behavior of theinput signal, as shall be discussed in more detail below, in “AdaptiveSnap-Out”.

In a step 525, between time t0 and t1, all three comparators areblanked, and the outputs are allowed to settle to the desired precision.During this settling period, the outputs are not considered reliable,and therefore not used.

In a step 530, at a time t1, after the defined blanking period, outputsof all three comparators are read. This is also called the “polling”step.

In step 540, and also which at time t1, it is determined which two ofthe three comparators are closest to the input voltage levels.

In a step 545, these two closest comparators then become the new outer(top and bottom) comparators for the next sampling process.

In a step 550, also at time t1, the remaining comparator is assigned areference voltage level that bisects, or otherwise lay somewhere inbetween, the new top and bottom rail voltage levels.

In a step 560, the newly-defined outer comparators are deemed ready tosample (i.e., are “active” between times t1 and t2). However, duringthis time, the inner comparator whose reference has been changed at timet1 is inactive, and allowed to settle.

In a step 570, between t1 and t2, the asynchronous ADC 300 is checkingwhether the analog input signal crosses one of the outer railcomparators. If not, the method 500 advances to step 575. If so, themethod 550 advances to step 580.

There is a possibility that the input signal continues to lie betweentwo adjacent DAC levels for an extended time period and henceasynchronous samples would not be created by the process defined above.In such situations, as is illustrated in step 575, the ADC 300 tests fora “timeout” wherein it is determined if a specified time has passedbeyond which a timeout period would have occurred. If a time out has notoccurred, step 575 returns to step 530. If it has occurred, step 575goes to step 590.

In a step 580, at time t3, the input voltage level crosses the referenceof one of the active comparators, thereby producing an asynchronoussample. Note that the asynchronous sample is a 2-tuple having voltagelevel and a time stamp. The voltage level is equal to the referencelevel which was crossed by the input signal while the timestampcorresponds to the time at which the crossover takes places. In reality,the time stamp corresponds to the quantized value of the time. Step 580then returns to step 510.

In a step 590, there is a forced sample of output samples, such as witha voltage levels in-between [such as midway] the two DAC values.

In a further aspect of the method 500, an intermediate or in betweenreference value can be further elegantly extended to use n comparators,wherein “n” is greater than three, and wherein two of the comparatorsremain as the outer comparators, and [n−2] of them become the innercomparators.

Should this occur, in every polling period, [n−2] comparators becomeinner comparators and are assigned values in between the upper and lowercomparators.

Note that as an asynchronous converter computing new reference levelsusing the snap-out values, we may create levels that are hypotheticallybeyond the allowed range. Should this occur, the new reference levelsneed to be created, by saturating the reference level to the maximumextremities of the allowed range. In other words, the range is extendedonly to the voltage rails, but not beyond.

For example, if the DAC level span the ranges 0-4095, and the lastsample is created at DAC level 4000 with a snap out of 256, the newouter levels would be created as 4000-256=3744, and 4095. Now, shouldthis occur, in the three converter case, the inner comparator which wasprogrammed at the mid-point and would sit at 3920.

Adaptive Snap-Out

The method 500 discussed above uses three or more comparators with theirown reference level. Each time a sample is created, the outer-mostcomparators are widened back to predefined positions; as mentionedearlier, the widening is called “snap-out”. In other words, if thetrigger comparator had a reference DAC level L, after sample creationthe outer comparators have levels L−Δ, L+Δ respectively, wherein Δ thesnapout value.

This application now introduces the concept in greater detail of“Adaptive snapout” can be generally defined wherein the extent of thesnapout is adapted to the signal characteristics. This can mean that thewidening does not occur to the maximum range allowable (i.e. the rails),but can be, for example, “adaptively snapped back” to a smaller rangewithin the rails. The value of snap out Δ is adapted over time, in orderto improve the sampling rate. The adaptive snap out can be embodied, forexample, in the sample rate control unit 320.

In contrast to the sampling schemes described in the previous section ofmethod 500, the snap out value can be adaptively varied to track theinput signal characteristics, thereby increasing the average samplingrate when compared to the sampling rate obtained by using fixed snapoutvalue in the baseline adaptation scheme described above. The next fewsections describe various adaptation techniques.

In a further aspect, the “snap-out” could use a “slow adaptation” whichtracks the macro-characteristics of the signal, e.g., amplitude,frequency, etc. In addition, the ADC 300 could be using a “fastadaptation” which tracks the micro characteristics of the signal, suchas local quiescence periods, or peaks.

Turning to FIG. 6A, illustrated is an example of the asynchronous ADC300 constructed according to the principles of the present application.

As is illustrated, the asynchronous sampler 304 includes DACs 601,coupled comparators 603, and the ADC control 605, which controls theDACs 601. Generally, the ADC control 605 dynamically adjusts the DACreference thresholds so as to control snapback values. Please note thatthe asynchronous ADC converter includes the clock 306 and the resampler308.

FIG. 6B illustrates the ADC control 605 in more detail. The ADC control605 includes a snapback selector 625.

Comparators 603 are dynamically assigned various voltage referencevalues, as discussed in method 500. The method 500 can be controlled bythe ADC controller 605. The ADC control 605 includes a fixed snapoutpredictor 627, a slow predictor—snapout calculator 630, a fastpredictor—snapout calculator 640, an approximate slope predictor snapoutcalculator 650, and a predictor based approach snapout calculator 660.

The fixed snapout calculator 627 implements snapback without dynamicallychanging the snapback range

Slow Adaptation: Histogram Method

In the approach implemented in the slow predictor 630, first ahistogram, and then the cumulative distributive function CDF, F_(x)(x)of the voltage difference between successive asynchronous samples arecomputed.

In other words, as is illustrated by the method 700, first, in a step710 a plurality asynchronous 2-tuples are measured and stored for eachof the asynchronously sampled inputs.

Then, the voltage measurement of each of these 2-tuples is extracted ina step 720.

Then, the voltage difference is between those consecutive asynchronouslysampled point is determined in step 730.

Then, a histogram of this information is created in a step 740.

And then the cumulative distributive function CDF, F_(x)(x) of thevoltage difference between successive asynchronous samples is computedin a step 750.

In a step 760, a user selects a confidence range of the snapout value,

In a step 770, the snapout value is employed with the asynchronous ADCconverter to define an adaptive snapout range after a sampling hasoccurred. This snapout range may often be less than a fully allowablesnapout range.

Generally, the snapout value Δ_(slow) is taken as the value which coversa large portion (say 99%) of the histogram, as represented by the CDF.That is, F_(x)(x<Δ_(slow))=0.99, where F_(x)(x) denotes the cumulative(probability) distribution function (CDF). The histogram is computedonly after a large number of asynchronous samples are accumulated, so asto ensure that they represent the long-term signal statisticsaccurately. Since the CDF is relatively stable, the value of Δ_(slow) isonly updated only infrequently. Hence, such an adaptation scheme cantrack slowly changing/unknown parameters like signal bandwidth or signalamplitude.

Fast Adaptation—Scaled Previous Snapout Approach

As will be described in the method 800 of FIG. 8, apart from the value(Δ_(slow)) from the slow adaptation loop as described above, the valueof snap out Δ can be changed based on local statistics (likeinstantaneous signal slope, previous snap out value, etc). This isreferred as Δ_(fast) to distinguish from the slow adaptation valueΔ_(slow),

If such adaptation can be done accurately and aggressively from sampleto sample, the output sampling rate can be increased. Employing such a“fast adaption” scheme will be given below, as employed in the fastpredictor snapout calculator 640.

As discussed above, after every sample, there is a blanking period afterwhich the two outermost reference levels become active. From the end ofthe initial blanking period till the time that the next sample isgenerated, the method 500 algorithm ensures that the reference levelsare reduced over time without ever losing the signal.

However if an “adaptive snapout scheme” is too aggressive, there ispossibility that the signal escapes during the initial blanking perioditself (i.e., the analog input signal is not bounded by the outercomparators.)

Hence, the snapout adaptation algorithms needs to strike a balancebetween the number of escape conditions (since signal moves faster whencompared to the range covered by the rail comparators due to smallersnapout) and the number of level crossing samples (which reduces whenthe snapout is too large). On the extreme, when the snapout values aretoo large (say rail voltage of the input signal), time outs may occurwhich reduces the average sampling rate.

The following equation gives one approach for calculating the snapoutafter the nth sample that represents this balance. Assuming thatΔ_(slow) is a power of “2”,Δ_(fast) ^(n+1)=2^(┌log) ² ^((min(max(k) ¹ ^(p,k) ² ^(),lΔ) ^(slow)^())┐),where p=maximum difference (in LSBs) between successive reference levelswhen the sample is created. Note that the reference levels may bedistributed non-uniformly for a given sample at a given time, and hencethe maximum difference between levels is used for p. AlsolΔ _(slow)

log₂(Δ_(slow)); k ₁ ,k ₂>=1.Note that, the parameters k₁, p and k₂ can be chosen to provide a goodtradeoff between maximizing the sampling rate and minimizing the signalescape probability. Note that Δ_(fast) ^(n) is bounded between k₂ andΔ_(slow).

Using this scheme, slow-moving signals (p ≃1), initially generatesamples at a slower rate resulting in smaller snapouts which in turnincrease the sampling rate. In contrast the faster moving sections ofthe signal initially produce larger values of p and faster samples, butthe resulting larger snap outs, result in lower sampling rates butminimize the probability of losing the signal. Thus this scheme adaptsdifferently to the fast and slow moving sections of the signal.

This can also be expressed in the method 800.

In a step 805, a plurality of asynchronous samples are generated.

In step 807 the largest number of Least Significant Bits (LSBs) betweentwo reference levels of the past polling cycle [“referred to as ‘p’].

In a step 810, assign values to escape variables k₁ and k₂ which denotea degree of tradeoff between escape probability and convergence time;

In a step 815, determine k₁ multiplied by p (now referred to as “q”.)

In a step 820, it is determined what is the maximum value q and k₂(referred to as “t”);

In a step 830, it is determined what the logarithm of this value (“t”)is in base 2. This value is called “ft”.

In a step 840, the method 800 selects the minimum value between thiscalculated value and the logarithm of “a slow” snapback values (e.g.,the output of method 700) (the minimum value referred to as “mv”).

In a step 850, after the minimum value is selected, the Ceil function of“mv” is determined (referred to as “mvceil”).

In a step 860, the value of step 850 is used as the “fast” snapout valueto set up for the next sample collection.

In a further aspect, whenever the signal escapes during the blankingperiod, it can be concluded that the signal is trapped between thepositive/negative rail and one of the outermost reference levels, say R.

In such a case, one option is to move the other outer comparator is tothe positive/negative rail so as to trap the signal with certainly.

However, a more aggressive option is to instead move the outercomparator some distance between R and the rail, such as moving it toR±2Δ_(slow), with the sign being determined based on whether the signalis above or below R.

Fast Adaptation—Approximate Slope Approach

In the approximate slope predictor 350, of the sampler 304 the nextsnapout value Δ_(fast) is decided based on the approximate slope of theinput signal at the current level crossing sample instant. That is,consider the first order approximation of the derivative of the signalat the current level crossing sample for one of the various scenarios inwhich the input signal crosses the level which is above the currentlevel.

$\begin{matrix}{\frac{\mathbb{d}V}{\mathbb{d}t} = \frac{{x\left( t_{n - 1} \right)} - {x\left( t_{n} \right)}}{{\tau\left( t_{n - 1} \right)} - {\tau\left( t_{n} \right)}}} \\{= \frac{{x\left( t_{n - 1} \right)} - \left( {{x\left( t_{n - 1} \right)} + {{snapout}_{n - 1}/k^{\lceil\frac{{\tau{(t_{n})}} - {\tau{(t_{n - 1})}}}{t_{poll}}\rceil}}} \right)}{{\tau\left( t_{n - 1} \right)} - {\tau\left( t_{n} \right)}}} \\{= \frac{{- {snapout}_{n - 1}}/k^{\lceil\frac{{\tau{(t_{n})}} - {\tau{(t_{n - 1})}}}{t_{poll}}\rceil}}{{\tau\left( t_{n - 1} \right)} - {\tau\left( t_{n} \right)}}} \\{= \frac{{snapout}_{n - 1}}{k^{\lceil\frac{{\tau{(t_{n})}} - {\tau{(t_{n - 1})}}}{t_{poll}}\rceil}\left\lbrack {{\tau\left( t_{n} \right)} - {\tau\left( t_{n - 1} \right)}} \right\rbrack}}\end{matrix}$where k is greater than or equal to one, and where T_(poll) is the timebetween two consecutive polling time instants.

Generally, although the input signal can cross the same reference levelat different time intervals depending on its nature, in general it istrue that high slew rate signals cross the immediate levels within shorttime intervals than the low slew rate signals. This is exploited bysetting the next snapout value proportional to the difference betweenthe reference values of rail comparators. That is, if the signal isslow, then it takes more time to cross the levels and hence thedifference between the reference values is also small.

On the other hand, for fast moving signals, it crosses the levels withinfew polling time intervals and hence the difference between thereference values of the rail comparators is large.

Fast Adaptation—Predictor Based Approach

In the fast snapout methods the improvement in the output rate of theasynchronous samples are in the same order since both are based on theapproximation of the slope of the signal at the current level crossingsample.

However, to increase this rate further one can employ predictivealgorithm to guess where the signal will be based on the past outputsamples. In this method the next snap out value is calculated based onthe output of a predictor at time δ_(n+1) after the last polling time,as is illustrated in the predictor. As one of the embodiment, a secondorder predictor is described whose input is the last two asynchronoussamples.

This is illustrated in FIG. 9, as embodied of the predictor basedapproach calculator 660. Depending on the next predicted sample, valuewe set the snapout in order to maximize the sampling rate and minimizethe signal escaping probability.

The predictor calculator 660 includes an input buffer 954, which thenconveys the asynchronous samples to a localized frequency estimator 956.The output of the localized frequency estimator 956 is then conveyed toa coupled predictor coefficient computation block 958. The output ofthis coefficient computation block 958 is then conveyed to the coupled2^(nd) order predictor 962. Also, please note that the asynchronoussamples are also conveyed in parallel to a buffer 960, which are thenalso conveyed to the 2^(nd) order predictor 962.

From the 2^(nd) order predictor 962, the predicted values are thenconveyed to the next snapout computation block 964. The next snapoutcomputation block 964 also receives past snapout values from the pastsnapout memory 968, and past samples from the buffer 960. Once thecalculation is made, the next snapout value is conveyed to the DACs andalso stored in the past snapout memory 968.

The localized frequency of the signal in the local frequency estimator956 is estimated by computing the average of dV/dT from the pastsamples. Since the sampling instants are non-uniformly spaced, thisestimate is smoothened using a low pass filter. In one of theembodiments, estimated local frequency is computed as

${\hat{f}}_{n} = {{\left( {1 - \alpha} \right){\hat{f}}_{n - 1}} + {\alpha\left( \frac{\left\lbrack \frac{\mathbb{d}V}{\mathbb{d}T} \right\rbrack_{n}}{4{V}_{\max}} \right)}}$where α is a parameter controlling the window over which exponentialaveraging is done, dV/dT is the average slew rate of the signal computedas

${\left\lbrack \frac{\mathbb{d}V}{\mathbb{d}T} \right\rbrack_{n + 1} = {{\left( {1 - \gamma} \right)\left\lbrack \frac{\mathbb{d}V}{\mathbb{d}T} \right\rbrack}_{n} + {\gamma\left\lbrack \frac{x_{n + 1} - x_{n}}{\tau_{n + 1} - \tau_{n}} \right\rbrack}}},$where γ is a parameter controlling the window over which exponentialaveraging is done.

The predictor coefficients are calculated asc ₁=2r cos(2π{circumflex over (f)} _(n) ΔT _(s)(n)), c ₂ =r ²,where r=0.95 is assumed, ΔTs(n) is the average time difference betweentwo adjacent samples computed asΔT _(s)(n+1)=(1−β)ΔT _(s)(n)+β[τ_(n+1)−τ_(n)],where β is a parameter controlling the window over which exponentialaveraging is done. The predictor 660 output is given by{circumflex over (x)} _(n+1) =−c ₁ x _(n) +c ₂ x _(n−1).

The snapout value computed and stored in buffer 960 as

${\Delta_{fast}^{n + 1} = {\min\left( {\frac{\Delta_{fast}^{n}}{K},\left\lfloor {{{{\hat{x}}_{n} - x_{n}}}2^{B}} \right\rfloor} \right)}},$where 2^(B) denotes the number of DAC levels supported and └x┘ denotesfloor operation and |x| denotes the absolute value and K is aprogrammable value.

Reconstruction Algorithms

FIG. 10A illustrates more details of the digital reconstruction andresampler 308. It includes an Akima calculator for reconstructingsynchronous samples 325, and a modified Akima calculator 330, and anadvanced resconstructor, such as a least squares solver, forreconstructing asynchronous samples 340.

The resampler 308 includes a reconstruction method selector 1005. Thereconstruction method selector 1005 selects from the algorithms 325,330, 340 to select a reconstruction method to generate synchronoussamples from asynchronous 2-tuples. Please note that synchronous samplesfrom the Akima calculator 325 or the modified Akima calculator 330 maybe fed into the advanced reconstructor 340.

Turning now to FIG. 10B, illustrated is an example baseline Akimaalgorithm and its modified version to be used for reconstruction of asynchronous sample in the digital reconstruction and resampler 308 usedwithin the reconstructors 325, 330.

The objective of these algorithms, the Akima algorithm and the modifiedAkima algorithm, is to reconstruct the original signal, given a set ofnon-uniform samples from a band-limited signal.

It is known that sinc-kernel based reconstruction can be used toperfectly reconstruct the signal, when the average sampling rate ishigher than the Nyquist rate. However such sinc-kernel basedreconstruction is computationally infeasible. One way to tradeoffperformance with complexity in this scheme is by using truncated sinckernels, but the method remains computationally expensive.

In the prior art, spline interpolation is another alternative algorithmwhich allows an attractive performance complexity tradeoff. Improvedperformance can be obtained, at the cost of higher complexity, by usinghigher order splines or by increasing the input sampling rate.

However, as understood by the present inventors, at reasonable inputsampling rates, the Akima algorithm achieves moderate performance at lowcomplexity. In fact, for moderate performance requirements, the Akimaalgorithm is lower complexity than truncated sinc reconstruction or evenspline interpolation. Moreover, the complexity of the Akima algorithmdepends on the output sampling rate and is independent of the inputsampling rate, whereas the complexity of the other algorithms aredependent on both the input and output sampling rates.

Using FIG. 10B, the baseline Akima algorithm is explained here. It fitsa (different) third order piecewise polynomial function to every regioncontaining an output point using the asynchronous samples created by theasynchronous ADC 330. Please note that the modified Akima algorithm isalso illustrated here in this figure, as shall be detailed below.

For baseline Akima calculator, assume that a sequence of input(asynchronous or synchronous) points are available, and the six pointsaround (three on either side) the first (or second) output point aredenoted as A₁₋₁, A₁₋₂, . . . , A₁₋₆ (or A₂₋₁, A₂₋₂, . . . , A₂₋₆,).Other asynchronous points, if any, are naturally punctured, i.e.,disposed of Let each of the points of interest, be represented by a2-tuple (x_(i,j), y_(i,j)), where i refers to the output index and j=1,. . . , 6 represent the six asynchronous sample indices around it. Thesesix points are used to completely describe the polynomial between A₁₋₃and A₁₋₄ (the two asynchronous points surrounding the output point) Thepiece-wise polynomial can be expressed as,y(x)=p ₀ +p ₁(x−x _(i,3))+p ₂(x−x _(i,3))+p ₃(x−x _(i,3))³,so that it satisfies the following conditions,

y(x_(i, 3)) = y_(i, 3); y(x_(i, 4)) = y_(i, 4);${\frac{\mathbb{d}y}{\mathbb{d}x}❘_{x = x_{i,3}}} = l_{3}$ and${\frac{\mathbb{d}y}{\mathbb{d}x}❘_{x = x_{i,4}}} = l_{4}$where l₃ and l₄ are the slopes of the polynomial at A₁₋₃ and A₁₋₄respectively.

Solving these equations, the following coefficient values are obtained.

p₀ = y_(i, 3); p₁ = l₃;${p_{2} = {\left( {{3\frac{\left( {y_{i,4} - y_{i,3}} \right)}{\left( {x_{i,4} - x_{i,3}} \right)}} - {2\; l_{3}} - l_{4}} \right)/\left( {x_{i,4} - x_{i,3}} \right)}};$$p_{3} = {\left( {l_{3} + l_{4} - {2\frac{\left( {y_{i,4} - y_{i,3}} \right)}{\left( {x_{i,4} - x_{i,3}} \right)}}} \right)/{\left( {x_{i,4} - x_{i,3}} \right)^{2}.}}$

Note that, the slopes of polynomial at the two endpoints, (x_(i,3),y_(i,3)) and (x_(i,4), y_(i,4)), are given as

$l_{3} = \frac{{{{m_{4} - m_{3}}}m_{2}} + {{{m_{2} - m_{1}}}m_{3}}}{{{m_{4} - m_{3}}} + {{m_{2} - m_{1}}}}$and${l_{4} = \frac{{{{m_{5} - m_{4}}}m_{3}} + {{{m_{3} - m_{2}}}m_{4}}}{{{m_{5} - m_{4}}} + {{m_{3} - m_{2}}}}},{where}$$m_{j} = {\frac{y_{i,{j + 1}} - y_{i,j}}{x_{i,{j + 1}} - x_{i,j}}.}$

A straightforward implementation of this algorithm, assuming alldivisions are implemented as look-up tables, would need 17 multipliersand 20 adders (ignoring multipliers with constants and table-lookups).

To restate the above, as implemented in the Akima calculator 325 of theADC 300, the Akima calculator 325 can generate synchronous voltagesamples of the ADC converters according to the flowchart, illustrated bymethod 1100, as given in FIG. 11.

In a step 1110, select three 2-tuples before and three after a selectedsynchronous point.

In a step 1120, assume a third order polynomial between the two closestasynchronous sample points surrounding the selected synchronous samplepoint.

In a step 1130, calculate the coefficients of the third order polynomialbased on the value of the previous time asynchronous sample, the timedifference between the asynchronous samples surrounding the selectedsample, and the five linear slopes of the line segments between thethree points before and the points after the selected synchronous samplepoint, including the slope of the selected point.

In a step 1140, evaluate the third order polynomial at the synchronoustime instant of interest.

In a step 1150, generate the synchronous ADC value based on thiscalculation.

In a step 1160, use the ADC value as the desired voltage level of thesynchronous sample, wherein the synchronous ADC value is generated basedon this calculation.

Modified Akima Algorithm

As advantageously appreciated by the present inventors, modificationsdone to the baseline Akima algorithm within another aspect of theasynchronous ADC 300.

The modified version of the algorithm uses four input points (unlike thebaseline algorithm which uses six points) around each output point. Forexample, points A₁₋₂, . . . , A₁₋₅ (also denoted as M₁₋₁, M₁₋₂, . . . ,M₁₋₄) are used for the first reconstructed point and points A₂₋₂, . . .A₂₋₅ (or M₂₋₁, . . . M₂₋₄) are used for the second output point. Otherasynchronous points, if any, are dropped (punctured) as before.

Apart from using fewer points, various other simplifications were madeto the baseline algorithm, resulting in the two versions of thealgorithm generating different cubic polynomials and estimatedsynchronous points.

The exact nature of the changes made to the baseline algorithm issummarized below.

The slope of the polynomial at point A₀, l₃, can be written as

${l_{3} = \frac{{w_{1}m_{2}} + {w_{2}m_{3}}}{w_{1} + w_{2}}},$where w₁=|m₄−m₃| and w₂=|m₂−m₁|. In this work, the weights are modifiedas modified as w₁=|x_(i,3)|^(n) and w₂=|x_(i,3) x_(i,2)|^(n), where n isan programmable real number, typically 1≦n≦2. The equations for l₄ canalso be modified similarly. These modifications are intuitive in thesense that the point closer to the point of interest, gets a much higherweight.

Also as per this modification, only two neighboring points are needed tocalculate the slope of the polynomial at each point, as opposed to fourneighboring points. Note that, by this modification two multipliers (foreach output point) are saved.

Instead of assuming the curve to be expressed as the third orderpolynomial given in baseline Akima algorithm outlined in the previoussection, the following equation may be used:y(x)=p ₀ +p ₁(x−x _((i,3)))+p ₂(x−x _((i,3)))(x−x _((i,4)))+p ₃(x−x_((i,3)))²(x−x _((i,4)))Putting the curve in the above form is better for fixed pointconsiderations, since higher powers of the difference on one end of thecurve are coupled with differences on the other end, thereby reducingoverall precision requirements in fixed point. The complexity can befurther reduced if the equations are normalized with respect to (x₃−x₂).Note that by doing this, the weight w₂=1.

Solving these equations, the following coefficient values are obtained,p ₀ =y _(i,2) ; p ₁ =m _(i,3) ; p ₂=(m ₃ −l ₃); p ₃=(l ₃ +l ₄−2m ₃)

For the coefficients, the following quantities are needed,

${l_{3}m_{3}} = {\left( {l_{3} - m_{3}} \right) = \frac{w_{2}\left( {m_{2} - m_{3}} \right)}{w_{1} + w_{2}}}$and${l_{4}m_{3}} = {\left( {l_{4} - m_{3}} \right) = {\frac{w_{2}\left( {m_{4} - m_{3}} \right)}{w_{2} + w_{3}}.}}$The last two coefficients can be rewritten as p₂=−l₃m₃; p₃=(l₃m₃+l₄m₃).It was also seen that performance can be marginallyimproved if:

${l_{3}m_{3}} = \frac{w_{2}\left( {{k_{1}m_{2}} - m_{3}} \right)}{w_{1} + w_{2}}$and${l_{4}m_{3}} = \frac{w_{2}\left( {{k_{2}m_{4}} - m_{3}} \right)}{w_{2} + w_{3}}$where k₁,k₂ are constants. Note that since, after normalization, w₂=1,l₃m₃ (and l₄m₃) can be calculated using only 1 addition and 1multiplication in addition to a table-lookup (ignoring themultiplication with a constant).

Turning now to FIG. 12, illustrated is a method 1200 that can beembodied in the modified Akima 330, using the modified Akima asdescribed above.

In a step 1210, select two 2-tuples before and two after a selectedsynchronous sample point.

In a step 1220, assume a third order polynomial between the two closestasynchronous sample points surrounding the selected synchronous samplepoint.

In a step 1230, calculate the coefficients of the third order polynomialbased on the value of the previous time asynchronous sample, the timedifferences between each of the asynchronous samples surrounding theselected sample, and the three linear slopes of the line segmentsbetween the two points before and the points after the selectedsynchronous sample point including over the selected point.

In a step 1240, the third order polynomial is evaluated at thesynchronous time instant.

In a step 1250, the synchronous ADC value is generated based on thiscalculation.

1260, this ADC value is used as the desired voltage level of thesynchronous sample to generate synchronous samples.

FIG. 13 a table that illustrates the numbers of adders and multipliersthat can implement the Akima and modified Akima algorithms. Forreference, the number of adders and multipliers that can implement thecubic spline algorithm is also shown. As is illustrated, the modifiedAkima may be implemented with less adders and multipliers than theAkima. Moreover, modified Akima has SNR performance which is very close(49 dB vs. 51 dB) to the cubic spline algorithm. In addition, the Akimaalgorithm is more hardware-friendly than the cubic spline, which stillrequires more adders and multipliers to implement.

FIG. 14A illustrates a high level description of a system 1400 of aleast squares solver for generating synchronous samples fromasynchronous samples.

As is illustrated, a “coarse reconstructor” 1410, such as an Akimareconstructor, receives asynchronous samples in 2-tuple form (x_(i),t_(i)) and converts them into synchronous samples (y_(m).), where y isthe voltage value and m is the synchronous time index. These “coarse”values, as well as the original 2-tuples, are then fed to a leastsquares solver (LSS) 1420. The least square solver generates the “fine”synchronous values (z_(m)) using the coarse values, y_(m), the originalasynchronous samples and its own past outputs. The main least squaressolver 1420, produces better synchronous estimates, z_(m) at time-pointsn_(m), using the coarse estimates y_(m). Note that the Least SquaresSolver (LSS) 1420 works a few output points behind the Akima enginesince it uses Akima outputs, in addition to its own previous outputs.

FIG. 14B illustrates a further aspect of the above. Instead of thecoarse reconstructor block, it can use a block implementing the modifiedAkima algorithm, 1460, which uses the asynchronous 2-tuples to firstproduce the coarse estimates, y_(m), and then uses both of these alongwith the previous outputs from the LSS block, similar to 1400, toproduce the refined synchronous estimates z_(m).

From the sampling theorem, we know that a band-limited signal f(t) canbe uniquely reconstructed from the uniformly sampled values, f(nT_(s))at sampling freq F_(s)=1/T_(s). Note that the uniformly sampled sincfunctions are the basis functions for a discrete time domainrepresentation of a band-limited signal.

${f(t)} = {\sum\limits_{n = {- \infty}}^{\infty}\;{{f\left( {nT}_{S} \right)} \cdot {{sinc}\left( \frac{t - {nT}_{s}}{T_{s}} \right)}}}$This is the convolution of the uniformly sampled values of the function(also called the coefficients) and the sinc functions centered atuniformly distributed sample time-points, as shown in FIG. 15A.

In practice, to reduce complexity, only a finite number of side-lobes ofthe sinc functions are retained by windowing. In one of the embodiments,a Blackman-Harris windowed sinc with w=7 side-lobes on either side isused (stored in a Look-up-table) as shown in FIG. 15B. Hence, the valueof the signal at any asynchronous time-point t can be written in termsof the values at synchronous time-points on either side as given below,

${f(t)} = {\sum\limits_{n = {- w}}^{w}\;{{f\left( {nT}_{S} \right)} \cdot {{wsinc}\left( {t - {nTs}} \right)}}}$

Conventional “Block Solver” algorithms work on a block of asynchronousinputs based on the above equation to formulate a system of equations asgiven below.[X] _(px1) =[W] _(pxq) ·[Z] _(qx1)where X are the asynchronous observations at p time-points, W, are thewindowed-sinc functions & Z are the values at q synchronous time-pointsin the block.

This system of equations is solved in the least-squares sense by thepseudo-inverse computation:Z=W ⁺ ·X=(W ^(T) W)⁻¹ ·W ^(T) ·XThis prior art approach gives the outputs at all the synchronoustime-points in the entire block from which a few outputs on either endsare discarded to remove edge effects.

FIG. 16 shows the points used in estimating the signal value, z₀ at thesynchronous time point n₀.f(t ₀)=f(n _(−w))·wsinc(t ₀ −n _(−w))+ . . . +f(n ⁻¹)·wsinc(t ₀ −n⁻¹)+f(n ₀)·wsinc(t ₀ −n ₀)+f(n ₁)·wsinc(t ₀ −n ₁)+f(n ₂)·wsinc(t ₀ −n₂)+ . . . +f(n _(w))·wsinc(t ₀ −n _(w))Using the coarse estimate f(n_(i))=y_(i) for i>=2 and fine estimate forf(n_(i))=z_(i) for i<2, the above equation can be written as,x ₀ =z _(−w) wsinc(t ₀ −n _(w))+ . . . +z ⁻¹ wsinc(t ₀ −n ⁻¹)+z ₀wsinc(t ₀ −n ₀)+z ₁ wsinc(t ₀ −n ₁)+y ₂ wsinc(t ₀ −n ₂)+ . . . +y _(w)wsinc(t ₀ −n _(w))Similarly, the equation for the observation (x⁻¹,t⁻¹) can be written as:x ⁻¹ =z _(−w) wsinc(t ⁻¹ −n ⁻¹)+ . . . +z ⁻¹ wsinc(t ⁻¹ −n ⁻¹)+z ₀wsinc(t ⁻¹ −n ₀)+z ₁ wsinc(t ⁻¹ −n ₁)+y ₂ wsinc(t ⁻¹ −n ₂)+ . . . +y_(w) wsinc(t ⁻¹ −n _(w))All the x_(i), t_(i), n_(m), y_(m) & z_(m(m<0)) are known. By looking upthe windowed sinc values (wsinc) from lookup tables & evaluatingconstants, the following equations follow,c ₀ =z ₀ s ₀₀ +z ₁ s ₀₁ and c ⁻¹ =z ₀ s ⁻¹⁰ +z ₁ s ⁻¹¹wheres _(1m) =wsinc(t ₁ −n _(m))&c ₁ =x ₁ −z _(−w) wsinc(t ₁ −n _(−w))− . . . −z ⁻¹ wsinc(t ₁ −n ⁻¹)−y ₂wsinc(t ₁ −n ₂)− . . . −y _(w) wsinc(t ₁ −n _(w))From these, z₀ can easily be found asz ₀=(c ₀ s ⁻¹¹ −c ⁻¹ s ₀₁)/(s ₀₀ s ⁻¹¹ −s ⁻¹⁰ s ₀₁)Thus, in this simple embodiment, the least squares solution reduces to aclosed form expression. z1 is not calculated from this set of equationsgiven in Eqn. 8, as it would have a large error since the asynchronouspoints chosen are far away from it & the sinc function value will be lowat that distance.

After z₀ is found, the entire window is made to slide and center at n1.z₁ & z₂ are treated as unknowns, y_(w+1) is taken from Akima and theequations at two asynchronous points around n₁ are solved to yield z₁.The process is thus repeated at every output point.

In general, the LSS can be formulated to fit many asynchronousobservations (x_(i),t_(i)) around the output point, with several z_(m)'s(m>=0) kept unknown (y_(m)'s not used for these). This results in a setof matrix equations[C] _(px1) =[S] _(pxq) ·[Z] _(qx1),where p is the number of asynchronous observations to solve for & q isthe number of unknown synchronous outputs.

This can be solved by the pseudo-inverse computation:Z=S ⁺ ·C=(S ^(T) S)⁻¹ ·S ^(T) ·C.For small matrix sizes, it may be practical to get a closed-formexpression for the output z0. Increasing the window size, w gives betterperformance, and it could be used to reduce complexity by lowering p & q(and offset the performance loss due to this).

Contrast this with conventional formulations which solve simultaneousequations at all the asynchronous points in a block (usually needs apseudo-inverse computation of a large sized matrix).

In FIG. 17, a comparison of the complexity of the proposed algorithmwith other prior art is given. Operations are listed in terms ofmultiplications, additions & table look-ups at the input asynchronousrate, R_(A) and output synchronous rate, R_(s). For typical bit-widths,a multiply operation can be considered equivalent to 16 additionoperations, so this mostly determines the algorithm complexity and powerin a hardware implementation. Divisions are implemented as a look-uptable (LUT) followed by a multiplier. As an illustration, the number ofmultiply operations is shown for the case the asynchronous rate is 12×the largest bandwidth of the input signal (i.e. R_(A)=6R_(S) assumingthe output is sampled at the nyquist rate).

Note that the complexity of the three algorithms, namely the Akima, themodified Akima and the Augmented LSS scheme are dependent only on thesynchronous output rate (R_(s)), while the cubic spline and the blocksolver solutions depend on the asynchronous input rate (R_(A)) inaddition to the synchronous output rate. Assuming R_(A)=6 R_(s) as seenin typical applications, the complexity of the modified Akima is theleast, followed by the original Akima algorithm and then the cubicspline. The proposed scheme, augmented LSS, is similar in multiplicativecomplexity to the cubic spline, and is lower in additive complexity.

In addition, FIG. 18 shows the reconstruction performance of the variousalgorithms. It can be seen that original Akima algorithm has the lowestperformance, followed by the modified Akima and then the referenceAkima. But the performance difference between these three algorithms issmall, although their complexity difference, as seen from FIG. 17, islarge. Thus the Akima algorithm and its modified versions offerattractive performance-complexity tradeoffs compared to the referencespline algorithm. The performance—complexity tradeoff gets moreattractive for the proposed LSS algorithm as it provides significantlybetter performance at complexity comparable to the cubic splinealgorithm.

Block solver algorithms give very good results, but have a very highimplementation cost, so have not been considered in this comparison.

FIG. 19 illustrates a method 1900 of generating an improved synchronousvalue from the least squares solvers 1420 (FIG. 14A) and 1430 (FIG.14B).

In a step 1910, the coarse reconstruction 1410 or the modified Akimareconstruction 1460 receives a plurality of 2-tuples of asynchronouslysampled Inputs.

In a step 1920 a coarse asynchronous to synchronous conversion isperformed on the plurality of 2-tuples to generate a plurality of lowprecision synchronous outputs.

In a step 1930, a high precision synchronous output, z₀ is generated,and is conveyed as well as a plurality of two tuples and low precisionsynchronous outputs after it, and its own previous high precisionoutputs (from previous steps).

In a step 1940, c₀ and c⁻¹ are calculated by summing the future lowprecision outputs and the past high precision outputs, after they areweighted with the appropriate windowed sinc. values and then subtractedfrom appropriate asynchronous samples.

In a step 1950, the four quantities he four quantities “s⁻¹¹”, “s₀₁”,“s₀₀” and “s⁻¹⁰” are calculated based on particular values of thewindowed sinc. function.

In a step 1960, Using c₀, c⁻¹, s⁻¹¹, s₀₁, s₀₀ and s⁻¹⁰, the highprecision synchronous output of interest, z₀ is generated.

Advantages of Least Squares Solver Approach

As compared to traditional block solver based approaches (that solve asimultaneous set of equations at the asynchronous points & require amatrix inversion), the complexity & power consumption of a least squaressolver (LSS) reconstruction engine is heavily reduced.

A simple, non-iterative form, involving only forward computations makeit conducive to hardware/real-time implementations since pipelining theinternal computations does not impact the overall throughput [Note: bothSpline interpolation & block solvers suffer from this problem]

Low latency compared to block/matrix inversion based implementations, aswe don't have to wait for data of the whole block to reconstruct thefirst point. [Note: Akima algorithm is well suited for the augmentationas it also has a non-iterative form & low latency]

Those skilled in the art to which this application relates willappreciate that other and further additions, deletions, substitutionsand modifications may be made to the described embodiments.

What is claimed is:
 1. An apparatus, comprising: receiving a pluralityof 2-tuples of asynchronously sampled inputs at an asynchronous tosynchronous reconstructor; performing a coarse asynchronous tosynchronous conversion using the plurality of 2-tuples to generate aplurality of low precision synchronous outputs; generating a highprecision synchronous output, z₀, using a plurality of asynchronous2-tuples and low precision synchronous outputs after it, and its ownprevious high precision outputs (from previous steps); calculating c₀and c⁻¹ by summing future low precision outputs and the past highprecision outputs after they are weighted with the appropriate windowedsinc, values and then subtracted from appropriate asynchronous samples;calculating, the four quantities four quantities s⁻¹¹, s₀₁, s₀₀ and s⁻¹⁰based on particular values of the windowed sinc, function; and using c₀,c⁻¹, s⁻¹¹, s₀₁, s₀₀ and s⁻¹⁰ to generate the high precision synchronousoutput of interest, z₀.
 2. The method of claim 1, wherein the coarsereconstruction algorithm is an Akima algorithm.
 3. The method of claim1, wherein the coarse reconstruction algorithm is a modified Akimaalgorithm.
 4. An apparatus, comprising: means for receiving a pluralityof 2-tuples of asynchronously sampled inputs at an asynchronous tosynchronous reconstructor means for performing a coarse asynchronous tosynchronous conversion using the plurality of 2-tuples to generate aplurality of low precision synchronous outputs; means for generating ahigh precision synchronous output, z₀, using a plurality of asynchronous2-tuples and low precision synchronous outputs after it, and its ownprevious high precision outputs (from previous steps); means forcalculating c₀ and c⁻¹ by summing future low precision outputs and thepast high precision outputs after they are weighted with the appropriatewindowed sinc, values and then subtracted from appropriate asynchronoussamples; means for calculating, the four quantities four quantitiess⁻¹¹, s₀₁, s₀₀ and s⁻¹⁰ based on particular values of the windowed sinc,function; and means for using c₀, c⁻¹, s⁻¹¹, s₀₁, s₀₀ and s⁻¹⁰ togenerate the high precision synchronous output of interest z₀.
 5. Asystem, comprising: an asynchronous sampler, including: at least one ofmore digital to analog converter (DAC); at least three voltagecomparators, wherein the asynchronous sampler is configured to: receivean analog input; determine an upper outer rail and a lower outer rail aspolling values to be used by voltage comparators; blank at least threecomparators; determine which two of the at least three comparators areclosest to the input analog voltage levels; select the two comparatorswhich are closest to the analog input signal to be the next outercomparators of the next sampling process; assign at least one remainingcomparator at a voltage level in between the new top and bottom voltagelevels; enable the outer rails, but blanking the inner rail;progressively narrow down the voltage range spanned by the two outercomparators; and generate a value of an asynchronous voltage comparatorcrossing; the ADC further comprising a digital reconstructor configuredto: select two Two-Tuples before and two after a selected synchronousADC conversion point; calculate the coefficients of a third orderpolynomial based on the value of the previous time asynchronous sample,the time differences between each of the asynchronous samplessurrounding the selected sample, and the three linear slopes of the linesegments between the two points before and the points after the selectedsynchronous sample point, including the slope of the selected point;evaluate the third order polynomial at the synchronous time instant;generate the synchronous ADC value based on this calculation; and usethe ADC value as the desired voltage level of the synchronous sample,wherein the synchronous ADC value is generated based on thiscalculation; wherein this ADC value is then further employed with aleast squares solver, configured to: generate a high precisionsynchronous output, z₀, using a plurality of asynchronous 2-tuples andlow precision synchronous outputs after it, and its own previous highprecision outputs (from previous steps); calculate c₀ and c⁻¹ by summingfuture low precision outputs and the past high precision outputs afterthey are weighted with the appropriate windowed sinc, values and thensubtracted from appropriate asynchronous samples; calculate the fourquantities s⁻¹¹, s₀₁, s₀₀ and s⁻¹⁰ based on particular values of thewindowed sinc, function; and use c₀, c⁻¹, s⁻¹, s₀₁, s₀₀ and s⁻¹⁰, togenerate the high precision synchronous output of interest, z₀.